Schematic illustration of a spectral submanifold ${\displaystyle {\mathcal {W}}(E)}$ emanating from a spectral subspace ${\displaystyle E}$. A trajectory ${\displaystyle p(t)}$ in the reduced coordinates is mapped to the phase space via the manifold parametrization ${\displaystyle W(p)}$.^[1]

In dynamical systems, a spectral submanifold (SSM) is the unique smoothest invariant manifold serving as the nonlinear extension of a spectral subspace of a linear dynamical system under the addition of nonlinearities.^[2] SSM theory provides conditions for when invariant properties of eigenspaces of a linear dynamical system can be extended to a nonlinear system, and therefore motivates the use of SSMs in nonlinear dimensionality reduction.

Definition

Consider a nonlinear ordinary differential equation of the form

${\displaystyle {\frac {dx}{dt}}=Ax+f_{0}(x),\quad x\in \mathbb {R} ^{n},}$

with constant matrix ${\displaystyle \ A\in \mathbb {R} ^{n\times n}}$ and the nonlinearities contained in the smooth function ${\displaystyle f_{0}={\mathcal {O}}(|x|^{2})}$.

Assume that ${\displaystyle {\text{Re}}\lambda _{j}<0}$ for all eigenvalues ${\displaystyle \lambda _{j},\ j=1,\ldots ,n}$ of ${\displaystyle A}$, that is, the origin is an asymptotically stable fixed point. Now select a span ${\displaystyle E={\text{span}}\,\{v_{1}^{E},\ldots v_{m}^{E}\}}$ of ${\displaystyle m}$ eigenvectors ${\displaystyle v_{i}^{E}}$ of ${\displaystyle A}$. Then, the eigenspace ${\displaystyle E}$ is an invariant subspace of the linearized system

${\displaystyle {\frac {dx}{dt}}=Ax,\quad x\in \mathbb {R} ^{n}.}$

Under addition of the nonlinearity ${\displaystyle f_{0}}$ to the linear system, ${\displaystyle E}$ generally perturbs into infinitely many invariant manifolds. Among these invariant manifolds, the unique smoothest one is referred to as the spectral submanifold.

An equivalent result for unstable SSMs holds for ${\displaystyle {\text{Re}}\lambda _{j}>0}$.

Non-autonomous extension

The theory on spectral submanifolds extends to nonlinear non-autonomous systems of the form

${\displaystyle {\frac {dx}{dt}}=Ax+f_{0}(x)+\epsilon f_{1}(x,\Omega t),\quad \Omega \in \mathbb {T} ^{k},\ 0\leq \epsilon \ll 1,}$

with ${\displaystyle f_{1}:\mathbb {R} ^{n}\times \mathbb {T} ^{k}\to \mathbb {R} ^{n}}$ a quasiperiodic forcing term.^[4]

Significance

Spectral submanifolds are useful for rigorous nonlinear dimensionality reduction in dynamical systems. The reduction of a high-dimensional phase space to a lower-dimensional manifold can lead to major simplifications by allowing for an accurate description of the system's main asymptotic behaviour.^[5] For a known dynamical system, SSMs can be computed analytically by solving the invariance equations, and reduced models on SSMs may be employed for prediction of the response to forcing.^[6]

Furthermore these manifolds may also be extracted directly from trajectory data of a dynamical system with the use of machine learning algorithms.^[7]

See also

• Invariant manifold
• Nonlinear dimensionality reduction
• Lagrangian coherent structure

References

External links

• Tool for automated SSM computation